Abstract

A spatially varying refractive index leads to the bending of photon paths in a medium, which complicates the Monte Carlo modeling of a photon random walk. We show that the process of photon diffusion in a turbid medium with varying refractive index and curved photon paths can be mapped to the diffusion process in a medium with straight photon paths and modified optical properties. Specifically, the diffusion coefficient, the absorption, and the refractive index of the second medium should differ from the corresponding properties of the first medium by the factor of the squared refractive index of the first medium. The specific intensity of light in the second medium will then be equal to the specific intensity in the first medium divided by the same factor, which also means that the photon density distributions in the two media will be identical. In a Monte Carlo simulation the scaling property suggests that two different algorithms can be used to obtain the photon density distribution, namely, the algorithm with curved photon paths and given optical properties and the algorithm with straight photon paths and modified optical properties.

© 2007 Optical Society of America

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    [CrossRef] [PubMed]
  3. T. Khan and H. Jiang, "A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices," J. Opt. A, Pure Appl. Opt. 5, 137-141 (2003).
    [CrossRef]
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    [CrossRef]
  5. J.-M. Tualle and E. Tinet, "Derivation of the radiative transfer equation for scattering media with a spatially varying refractive index," Opt. Commun. 228, 33-38 (2003).
    [CrossRef]
  6. M. L. Shendeleva, "Radiative transfer in a turbid medium with a varying refractive index: comment," J. Opt. Soc. Am. A 21, 2464-2468 (2004).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  10. H. Dehghani, B. A. Brooksby, B. W. Pogue, and K. D. Paulsen, "Effects of refractive index on near-infrared tomography of the breast," Appl. Opt. 44, 1870-1878 (2005).
    [CrossRef] [PubMed]
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    [CrossRef]
  12. X. Liang, Q. Zhang, and H. Jiang, "Quantitative reconstruction of refractive index distribution and imaging of glucose concentration by using diffusing light," Appl. Opt. 45, 8360-8365 (2006).
    [CrossRef] [PubMed]
  13. L. Wang, S. Jacques, and L. Zheng, "MCML--Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
    [CrossRef] [PubMed]
  14. M. L. Shendeleva and J. A. Molloy, "Diffuse light propagation in a turbid medium with varying refractive index: Monte Carlo modeling in a spherically symmetrical geometry," Appl. Opt. 45, 7018-7025 (2006).
    [CrossRef] [PubMed]
  15. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergammon, 1970).

2006

2005

2004

2003

S. A. Alexandrov, A. V. Zvyagin, K. K. M. B. D. Silva, and D. D. Sampson, "Bifocal optical coherence refractometry of turbid media," Opt. Lett. 28, 117-199 (2003).
[CrossRef] [PubMed]

L. Martí -López, J. Bouza-Domínguez, J. C. Hebden, S. R. Arridge, and R. A. Martínez-Celorio, "Validity conditions for the radiative transfer equation," J. Opt. Soc. Am. A 20, 2046-2056 (2003).
[CrossRef]

A. V. Zvyagin, K. K. M. B. D. Silva, S. A. Alexandrov, T. R. Hillman, and J. J. Armstrong, "Refractive index tomography of turbid media by bifocal optical coherence refractometry," Opt. Express 11, 3503-3517 (2003).
[CrossRef] [PubMed]

T. Khan and H. Jiang, "A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices," J. Opt. A, Pure Appl. Opt. 5, 137-141 (2003).
[CrossRef]

J.-M. Tualle and E. Tinet, "Derivation of the radiative transfer equation for scattering media with a spatially varying refractive index," Opt. Commun. 228, 33-38 (2003).
[CrossRef]

H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, and K. D. Paulsen, "The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach," Phys. Med. Biol. 48, 2713-2727 (2003).
[CrossRef] [PubMed]

1995

L. Wang, S. Jacques, and L. Zheng, "MCML--Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

Alexandrov, S. A.

Armstrong, J. J.

Arridge, S. R.

Bal, G.

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergammon, 1970).

Bouza-Domínguez, J.

Brooksby, B.

H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, and K. D. Paulsen, "The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach," Phys. Med. Biol. 48, 2713-2727 (2003).
[CrossRef] [PubMed]

Brooksby, B. A.

Dehghani, H.

H. Dehghani, B. A. Brooksby, B. W. Pogue, and K. D. Paulsen, "Effects of refractive index on near-infrared tomography of the breast," Appl. Opt. 44, 1870-1878 (2005).
[CrossRef] [PubMed]

H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, and K. D. Paulsen, "The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach," Phys. Med. Biol. 48, 2713-2727 (2003).
[CrossRef] [PubMed]

Hebden, J. C.

Hillman, T. R.

Jacques, S.

L. Wang, S. Jacques, and L. Zheng, "MCML--Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

Jiang, H.

X. Liang, Q. Zhang, and H. Jiang, "Quantitative reconstruction of refractive index distribution and imaging of glucose concentration by using diffusing light," Appl. Opt. 45, 8360-8365 (2006).
[CrossRef] [PubMed]

T. Khan and H. Jiang, "A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices," J. Opt. A, Pure Appl. Opt. 5, 137-141 (2003).
[CrossRef]

Khan, T.

T. Khan and A. Thomas, "Inverse problem in refractive index based optical tomography," Inverse Probl. 22, 1121-1137 (2006).
[CrossRef]

T. Khan and H. Jiang, "A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices," J. Opt. A, Pure Appl. Opt. 5, 137-141 (2003).
[CrossRef]

Liang, X.

Lowery, A. J.

Martí -López, L.

Martínez-Celorio, R. A.

Molloy, J. A.

Paulsen, K. D.

H. Dehghani, B. A. Brooksby, B. W. Pogue, and K. D. Paulsen, "Effects of refractive index on near-infrared tomography of the breast," Appl. Opt. 44, 1870-1878 (2005).
[CrossRef] [PubMed]

H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, and K. D. Paulsen, "The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach," Phys. Med. Biol. 48, 2713-2727 (2003).
[CrossRef] [PubMed]

Pogue, B. W.

H. Dehghani, B. A. Brooksby, B. W. Pogue, and K. D. Paulsen, "Effects of refractive index on near-infrared tomography of the breast," Appl. Opt. 44, 1870-1878 (2005).
[CrossRef] [PubMed]

H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, and K. D. Paulsen, "The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach," Phys. Med. Biol. 48, 2713-2727 (2003).
[CrossRef] [PubMed]

Premaratne, E.

Premaratne, M.

Sampson, D. D.

Shendeleva, M. L.

Silva, K. K. M. B. D.

Silva, K. M. B. D.

Thomas, A.

T. Khan and A. Thomas, "Inverse problem in refractive index based optical tomography," Inverse Probl. 22, 1121-1137 (2006).
[CrossRef]

Tinet, E.

J.-M. Tualle and E. Tinet, "Derivation of the radiative transfer equation for scattering media with a spatially varying refractive index," Opt. Commun. 228, 33-38 (2003).
[CrossRef]

Tualle, J.-M.

J.-M. Tualle and E. Tinet, "Derivation of the radiative transfer equation for scattering media with a spatially varying refractive index," Opt. Commun. 228, 33-38 (2003).
[CrossRef]

Vishwanath, K.

H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, and K. D. Paulsen, "The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach," Phys. Med. Biol. 48, 2713-2727 (2003).
[CrossRef] [PubMed]

Wang, L.

L. Wang, S. Jacques, and L. Zheng, "MCML--Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergammon, 1970).

Zhang, Q.

Zheng, L.

L. Wang, S. Jacques, and L. Zheng, "MCML--Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

Zvyagin, A. V.

Appl. Opt.

Comput. Methods Programs Biomed.

L. Wang, S. Jacques, and L. Zheng, "MCML--Monte Carlo modeling of light transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

Inverse Probl.

T. Khan and A. Thomas, "Inverse problem in refractive index based optical tomography," Inverse Probl. 22, 1121-1137 (2006).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

T. Khan and H. Jiang, "A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices," J. Opt. A, Pure Appl. Opt. 5, 137-141 (2003).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

J.-M. Tualle and E. Tinet, "Derivation of the radiative transfer equation for scattering media with a spatially varying refractive index," Opt. Commun. 228, 33-38 (2003).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Med. Biol.

H. Dehghani, B. Brooksby, K. Vishwanath, B. W. Pogue, and K. D. Paulsen, "The effects of internal refractive index variation in near-infrared optical tomography: a finite element modelling approach," Phys. Med. Biol. 48, 2713-2727 (2003).
[CrossRef] [PubMed]

Other

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergammon, 1970).

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Figures (8)

Fig. 1
Fig. 1

Geometry of the model. The photon is emitted from the origin. The shadow indicates the varying refractive index of the medium.

Fig. 2
Fig. 2

Models considered in Example 1. (a) Optical properties of the model with curved photon paths. (b) Optical properties of the model with straight photon paths. β = 0 . Distance r from the origin is in cm. (c) Specific intensity ϕ (in W cm 2 ) for the model with curved photon paths at various times indicated in the figure. (d) Specific intensity ϕ ̂ (in W cm 2 ) for the model with straight photon paths at various times indicated in the figure. Black solid curves correspond to the analytical solutions; color curves show the results of the Monte Carlo simulations.

Fig. 3
Fig. 3

Models considered in Example 1. Photon densities ρ and ρ ̂ (in cm 3 ) for the models with curved photon paths and straight photon paths versus distance r from the origin (in cm) for various times indicated in the figure. Color curves show the results of the Monte Carlo simulations, and the black solid curves show the analytical solutions. Blue ( 0.5 ns ) , red ( 1 ns ) , and green ( 2 ns ) curves correspond to the curved-path model, and magenta ( 0.5 ns ) , light blue ( 1 ns ) , and yellow ( 2 ns ) curves correspond to the straight-path model.

Fig. 4
Fig. 4

Models considered in Example 2. (a) Optical properties of the model with curved photon paths. (b) Optical properties of the model with straight photon paths. β = 0 . Distance r from the origin is in cm. (c) Specific intensity ϕ (in W cm 2 ) for the model with curved photon paths at various times indicated in the figure. (d) Specific intensity ϕ ̂ (in W cm 2 ) for the model with straight photon paths at various curves indicated in the figure. Black solid curves correspond to the analytical solutions; color curves show the results of the Monte Carlo simulations.

Fig. 5
Fig. 5

Models considered in Example 2. Photon densities ρ and ρ ̂ (in cm 3 ) for the models with curved photon paths and straight photon paths versus distance r from the origin (in cm) for various times indicated in the figure. Color curves show the results of the Monte Carlo simulations, and the black solid curves show the analytical solutions. Blue ( 0.5 ns ) , red ( 1 ns ) and green ( 2 ns ) curves correspond to the curved-path model, and magenta ( 0.5 ns ) , light blue ( 1 ns ) , and yellow ( 2 ns ) curves correspond to the straight-path model.

Fig. 6
Fig. 6

Models considered in Example 3. (a) Optical properties of the model with curved photon paths. (b) Optical properties of the model with straight photon paths. β D b is in cm 2 , r in cm. (c) Specific intensity ϕ (in W cm 2 ) for the model with curved photon paths at various times indicated in the figure. (d) Specific intensity ϕ ̂ (in W cm 2 ) for the model with straight photon paths at various times indicated in the figure. Black solid curves correspond to the analytical solutions; color curves show the results of the Monte Carlo simulations.

Fig. 7
Fig. 7

Models considered in Example 3. Photon densities ρ and ρ ̂ (in cm 3 ) for the models with curved photon paths and straight photon paths versus distance r from the origin (in cm) for various times indicated in the figure. Color curves show the results of the Monte Carlo simulations, and the black solid curves show the analytical solutions. Blue ( 0.5 ns ) , red ( 1 ns ) and green ( 2 ns ) curves correspond to the curved-path model, and magenta ( 0.5 ns ) , light blue ( 1 ns ) and yellow ( 2 ns ) curves correspond to the straight-path model.

Fig. 8
Fig. 8

Absolute value of the relative deviation of the numerical photon density ρ num from analytical photon density ρ anal for the model shown in Fig. 7 for t = 0.5 ns . The red curve corresponds to the curved-path model, and the light blue curve corresponds to the straight-path model.

Equations (61)

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n 0 c ϕ t div [ D grad ( ϕ ) ] + β ϕ = 0 ,
n c ϕ t div [ D n 2 grad ( ϕ n 2 ) ] + β ϕ = 0 ,
D ̂ = D n 2 , β ̂ = β n 2 ,
ϕ ̂ = ϕ n 2 ,
n 3 c ϕ ̂ t div [ D ̂ grad ( ϕ ̂ ) ] + β ̂ ϕ ̂ = 0 .
n ̂ = n 3 , v ̂ = v n 2 .
n 3 c ϕ ̂ t 1 r 2 r ( D ̂ r 2 ϕ ̂ r ) = E 0 δ ( t ) δ ( r ) 4 π r 2 .
ψ ̂ = 0 ϕ ̂ exp ( p t ) d t ,
H = d r D ̂ r 2 ,
ψ ̂ = D ̂ r 4 n 3 p c ψ ̂ ,
ψ ̂ = g exp ( f p ) ,
g = g 0 H , f = f 0 H ,
n 3 D ̂ r 4 c = f 0 2 H 4 ,
ϕ ̂ = ϕ ̂ 0 2 π 1 2 t 3 2 exp ( f 0 2 4 t H 2 ) ,
0 n 3 c ϕ ̂ 4 π r 2 d r = E 0 ,
ϕ ̂ 0 = E 0 f 0 4 π .
H = h r ,
f 0 2 = 1 c D 0 3 n 0 3 .
ϕ ̂ = E 0 8 c 1 2 ( π n b D b t ) 3 2 exp ( r 2 4 c t D b 3 n b 3 h 2 ) .
ϕ = E 0 c n 2 8 ( π c t n b D b ) 3 2 exp ( r 2 4 c t D b 3 n b 3 h 2 ) .
H = 1 + exp ( a r ) D ̂ b r .
D ̂ = D ̂ b ( 1 + a r ) exp ( a r ) + 1 ,
n = n b { 1 + [ 1 + a r ] exp ( a r ) } 1 3 [ 1 + exp ( a r ) ] 4 3 .
D = D b [ 1 + exp ( a r ) ] 8 3 { [ 1 + a r ] exp ( a r ) + 1 } 5 3 .
ϕ = E 0 c n 2 8 ( π c t n b D b ) 3 2 exp { r 2 n b 4 c t D b [ 1 + exp ( a r ) ] 2 } .
H = [ 1 + exp ( a r ) ] α D ̂ b r ,
D ̂ = D ̂ b [ 1 + exp ( a r ) ] α 1 { 1 + exp ( a r ) [ 1 + α a r ] } ,
n = n b { 1 + exp ( a r ) [ 1 + α a r ] } 1 3 [ 1 + exp ( a r ) ] α + 1 3 .
ϕ = E 0 c n 2 8 ( π c t n b D b ) 3 2 exp { r 2 n b 4 c t D b [ 1 + exp ( a r ) ] 2 α } .
n = n b [ 1 + exp ( a r ) ] 2 3 { 1 + exp ( a r ) [ 1 a r ] } 1 3 ,
D = D b ( 1 + exp ( a r ) ) 2 3 { 1 + exp ( a r ) [ 1 a r ] } 5 3 .
ϕ = E 0 n 2 8 ( π t n b D b ) 3 2 c 1 2 exp { r 2 n b [ 1 + exp ( a r ) ] 2 4 c t D b } .
n 3 c ϕ ̂ t D ̂ 0 r 2 r ( r 2 ϕ ̂ r ) + β ̂ ϕ ̂ = E 0 δ ( t ) δ ( r ) 4 π r 2 .
d 2 U d r 2 = ( β ̂ D ̂ 0 + n 3 p c D ̂ 0 ) U .
W = U ̇ U ,
W ̇ + W 2 = β ̂ D ̂ 0 + n 3 c D ̂ 0 p ,
W = f p + g ,
f 2 = n 3 c D ̂ 0 ,
f ̇ + 2 f g = 0 ,
g ̇ + g 2 = β ̂ D ̂ 0 .
ψ ̂ = ψ ̂ 0 r exp ( p 0 r f d r + 0 r g d r ) ,
0 r g d r = ln ( f 0 f ) 1 2 ,
F = 0 r f d r ,
ϕ ̂ = ϕ ̂ 0 F 2 π 1 2 r t 3 2 ( n 0 n ) 3 4 exp ( F 2 4 t ) .
ϕ ̂ 0 = E 0 4 π D 0 n 0 2 .
ϕ ̂ = E 0 F 8 ( π t ) 3 2 r D 0 n 0 5 4 n 3 4 exp ( F 2 4 t ) .
ϕ = E 0 F 8 ( π t ) 3 2 r D 0 ( n n 0 ) 5 4 exp ( F 2 4 t ) .
g = a 1 + b exp ( ar ) ,
β ̂ = a 2 D ̂ 0 1 + b exp ( ar ) .
f = C 1 ( b exp ( ar ) 1 + b exp ( ar ) ) 2 ,
n = n b ( b exp ( ar ) 1 + b exp ( ar ) ) 4 3 ,
F = ( n b a 2 c D b ) 1 2 { b [ 1 exp ( ar ) ] [ 1 + b ] [ 1 + b exp ( ar ) ] + ln ( 1 + b exp ( ar ) 1 + b ) } .
ϕ = E 0 F b 8 ( π t ) 3 2 D b r ( 1 + b ) ( b exp ( ar ) 1 + b exp ( ar ) ) 5 3 exp ( F 2 4 t ) .
D = D b n b 2 n 2 ,
β = a 2 D b n b 2 n 2 [ 1 + b exp ( ar ) ] .
χ = exp ( μ s s ) ,
s = ln ( χ ) μ s ,
n r sin φ = C ,
μ ¯ s = 1 s 0 s μ s d s ,
0 s μ s d s = ln ( χ ) ,
ρ ̂ = ϕ ̂ n ̂ c ω = ϕ n c ω = ρ .

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